Geoscience Reference
In-Depth Information
pixel
u
, this latter marginal PDF is just a normalizing constant (a scalar). It is common to all
K
classes (i.e., it does not affect the allocation decision), and it is typically computed as
K
Â
1
*
*
*
p
[()]
xu
=
p
[()| ()
xu
c
u
= ◊
c
]
p
, to ensure that the sum of the resulting
K
conditional
k
k
k
=
pc
probabilities
{
[ ( ) |
uxu
( )],
k
= 1
,
…
,
K
}
is 1. The final step in the classification procedure is
k
typically the allocation of pixel
u
to the class
c
m
with the largest conditional probability:
*
*
pc
[ ( ) |
uxu
( )]
=
max{
pc
[ ( ) |
uxu
( )],
k
= 1
,
…
,
K
}
, which is termed
maximum a posteriori
(MAP)
m
k
k
selection.
In the case of Gaussian maximum likelihood (GML), the likelihood function is
B
-variate
Gaussian and fully specified in terms of the (B
1) class-conditional multivariate mean vector
and the (B
¥
¥
B) variance-covariance matrix
of reflectance values. The exact form
m
=
[{ ()|()
EX
u
c
u
=
c
},
b
=
1
, , ]'
…
B
k
b
k
S
k
=
[Cov{
XX c
( ),
u
( ) |
u
( )
u
=
c
},
b
=
1
,
º
,
Bb
, '
=
1
,
…
,
B
]
b
b
'
k
of the likelihood function then becomes:
(
)
-
12
/
==
()
◊
-
B
/
2
*
-
1
p
[()| ()
xu
c
u
c
]
2
p
S
◊ -
exp [()
xu
- ◊ ◊
m
]'
S
[()
xu
-
m
]/
2
(11.2)
k
k
k
k
k
where and denote, respectively, the determinant and inverse of the class-conditional
variance-covariance matrix .
In many cases, there exists ancillary information that is not accounted for in the classification
procedure by conventional classifiers. One approach to account for this ancillary information is
that of local prior probabilities, whereby the prior probabilities
-1
S
k
S
k
S
k
p
*
are replaced with, say, elevation-
*
[( )| ( )]
dependent probabilities
pc
uu
e
, where
denotes the elevation or slope value at pixel
e
()
u
k
u
. Such probabilities are location-dependent due to the spatial distribution of elevation or slope.
In the absence of ancillary information, the spatial correlation of each class (which can be
modeled from a representative set of training samples) provides important information that should
be accounted for in the classification procedure. Fragmented classifications, for example, might be
incompatible with the spatial correlation of classes inferred from the training pixels. This charac-
teristic can be expressed in probabilistic terms via the notion that a pixel
u
is more likely to be
classified in class
k
than in class
k'
, i.e.,
, if the information in the
pc
[()| ()]
uxu
>
p c
[()| ()]
uxu
k
k
'
neighborhood of that pixel indicates the presence of a
-class neighborhood. This notion of context
is typically incorporated in the remote sensing literature via Markov random field models (MRFs);
see, for example, Li (2001) or Tso and Mather (2001) for details.
k
11.2.2
Geostatistical Modeling of Context
In this chapter, we propose an alternative procedure for modeling context based on indicator
geostatistics, which provides another way for arriving at local prior probabilities
*
[( )|
given
pc
k
uc
]
g
the set of
class labels ; see, for example, Goovaerts (1997). Contrary
to the MRF approach, the geostatistical alternative: (1) does not rely on a formal parametric model,
(2) is much simpler to explain and implement in practice, (3) can incorporate complex spatial
correlation models that could also include large-scale (low-frequency) spatial variability, and (4)
provides a formal way of integrating other ancillary sources of information to yield more realistic
local prior probabilities.
Indicator geostatistics (Journel, 1983; Solow, 1986) is based on a simple, yet effective, measure
of spatial correlation: the covariance
G
c
=
[ (
c
u
),
g
=
1
,
…
,
G
]'
g
g
i
k
()
u
between any two indicators
and
of
s
k
()
h
i
k
(
uh
+
)
the same class separated by a distance vector
, and is defined as:
h
